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Sunday, 8 September 2013

Quantum Gates: Single-Qubit Gates

The axioms of quantum mechanics say that an isolated quantum system undergoes unitary evolution. Therefore, such operations are the natural means for governing the time evolution of the quantum computer. Whereas in the classical case ordinary Boolean logic gates are used to manipulate the bits, unitary transformations on the state space of the quantum computer are used to manipulate the qubits. In quantum computing these unitary transformations are also referred to as quantum gates. Due to the linearity of quantum mechanics, unitary operators have the ability to transform arbitrary superpositions of basis states. In this case, a quantum computer is essentially able to exploit an exponential degree of parallelism in its information processing tasks.

The vector space nature of a quantum system allows quantum gates to be represented by matrices by specifying the action on a particular choice of basis states. Therefore, the matrix representation will be a useful way to describe quantum operations.

Single Qubit Gates

The unitary \(U:\mathcal{H}^2\rightarrow\mathcal{H}^2\) operators that act on a single qubit can be represented as \(2\times2\) matrices called single qubit gates. An important set of 1-qubit quantum gates represented as a matrix in the computational basis
\[\left|0\right>=\begin{pmatrix}1\\0\end{pmatrix}, \left|1\right>=\begin{pmatrix}0\\1\end{pmatrix}\]
is the following set
\[I=\begin{pmatrix}1&0 \\ 0 &1\end{pmatrix},
X=\begin{pmatrix}0&1 \\ 1&0 \end{pmatrix},
Y=\begin{pmatrix}0&i \\ -i&0 \end{pmatrix},
Z=\begin{pmatrix}1&0 \\ 0&-1 \end{pmatrix}.\]
The acquainted reader might recognize these as the Pauli matrices. They all satisfy the properties that \(U^\dagger=U^{-1}\) for all \(U\in\{X,Y,Z\}\) and
\[\begin{align*}
XY&=-YX=iZ, \\
YZ &=-ZY=iX, \\
ZX &=-XZ=iY.
\end{align*}\]
A consequence of these relations is that any unitary operator \(U\) acting a single qubit can be represented in terms of the Pauli matrices as
\[U=c_II+c_Xx+c_YY+c_ZZ\]
for some real coefficients \(c_I, c_X, c_Y\) and \(c_Z\).

Another important quantum gate acting on a single qubit is the Hadamard gate \(H\). Its action can be given by specifying the way it acts on the basis states \(\left|0\right>\) and \(\left|1\right>\):
\[\begin{align*}
H\left|0\right> &=\frac{1}{\sqrt2}(\left|0\right>+\left|1\right>), \\
H\left|1\right> &= \frac{1}{\sqrt2}(\left|0\right>-\left|1\right>)
\end{align*}\]
The corresponding matrix representation for \(H\) is thus given by
\[H= \frac{1}{\sqrt{2}}\begin{pmatrix}
1&1 \\
1&-1 \\
\end{pmatrix}
\]
Using this matrix representation of \(H\) it is easily verified that \(H\) is indeed a unitary operator. In fact, since \(H=H^\dagger\), \(H\) can be regarded as its own inverse. The Hadamard gate \(H\) takes one of the basis states and creates a weighted superposition of the two basis states as shown.

The single qubit gates parameterized by an angle \(\theta\) defined as
\[R_\theta=\begin{pmatrix}
1& 0 \\
0&e^{i\theta}
\end{pmatrix}\]
is called the \(\theta\)-phase gate since
\[R_\theta(\alpha\left|0\right>+ \beta\left|1\right>)=\alpha\left|0\right>+e^{i\theta}\beta\left|1\right>,\]
and a general qubit \(\left|\psi\right>=\alpha\left|0\right>+\beta\left|1\right>\) can be thought of as picking up a relative phase factor of \(e^{i\theta}\).